The homological torsion of \(\mathrm{PSL}_2\) of the imaginary quadratic integers. (English) Zbl 1307.11065
Summary: The Bianchi groups are the groups \(\mathrm{(P)SL}_2\) over a ring of integers in an imaginary quadratic number field. We reveal a correspondence between the homological torsion of the Bianchi groups and new geometric invariants, which are effectively computable thanks to their action on hyperbolic space. We expose a novel technique, the torsion subcomplex reduction, to obtain these invariants. We use it to explicitly compute the integral group homology of the Bianchi groups. Furthermore, this correspondence facilitates the computation of the equivariant \(K\)-homology of the Bianchi groups. By the Baum/Connes conjecture, which is satisfied by the Bianchi groups, we obtain the \(K\)-theory of their reduced \(C^\ast\)-algebras in terms of isomorphic images of their equivariant \(K\)-homology.
MSC:
| 11F75 | Cohomology of arithmetic groups |
| 22E40 | Discrete subgroups of Lie groups |
| 57S30 | Discontinuous groups of transformations |
| 55N91 | Equivariant homology and cohomology in algebraic topology |
| 19L47 | Equivariant \(K\)-theory |
| 55R35 | Classifying spaces of groups and \(H\)-spaces in algebraic topology |
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